The Beckman-Quarles theorem for continuous mappings from R^2 to C^2
| dc.creator | Tyszka, Apoloniusz | |
| dc.date | 2002-06-25 | |
| dc.date.accessioned | 2026-07-07T04:49:21Z | |
| dc.date.available | 2026-07-07T04:49:21Z | |
| dc.description | Let ϕ((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:R^2 -> C^2 preserves distance d>=0 if for each x,y \in R^2 ϕ(x,y)=d^2 implies ϕ(f(x),f(y))=d^2. We prove that if x,y \in R^2 and |x-y|=(2\sqrt{2}/3)^k \cdot (\sqrt{3})^l (k,l are non-negative integers) then there exists a finite set {x,y} \subseteq S(x,y) \subseteq R^2 such that each unit-distance preserving mapping from S(x,y) to C^2 preserves the distance between x and y. It implies that each continuous map from R^2 to C^2 preserving unit distance preserves all distances. | |
| dc.description | 12 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/math/0206260 | |
| dc.identifier | http://arxiv.org/abs/math/0206260 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64388 | |
| dc.subject | Metric Geometry | |
| dc.subject | 51M05 (Primary) | |
| dc.title | The Beckman-Quarles theorem for continuous mappings from R^2 to C^2 | |
| dc.type | text |